EXAMPLES.

by

1. A ship from latitude 47° 30' N. has sailed S. W. What latitude is she in, and what departure

S. 98 miles.

has she made?

Let C be the place sailed from, CB the meridian, and BCA the course, which ,we find from the table of rhumbs to be equal to 33° 45'; then AC will be the distance sailed, equal to 98 miles. Also, AB will be the departure, and CB the difference of latitude.

Then by the formulas for the solution of right angled triangles,

As radius ar. c.

A

0.000000 As radius
9.919846: sin C 33° 45'
1.991226:: CA 98

ar. c.

0.000000

9.744739

1.991226

: CB

81.48

1.911072: AB 54.45

1.735965

Latitude left
Dif. lat.

81.48 miles = 81.48 minutes = 1° 22' S.

Departure, 54.45 miles.

2. A ship sails 24 hours on a direct course, from latitude 38° 32′ N. till she arrives at latitude 36° 56′ N. The course is between S. and E. and the rate 51⁄2 miles an hour. Required the course, distance, and departure.

Lat. left 38° 32′ N.

In lat. 36° 56'

24 × 51 132 miles = distance.

Diff. 1° 36' = = 96 miles.

=

As dist. 132 ar. c. 7.879426 | As radius ar. c. 0.000000

Hence, the course is S. 43° 20' E., and the departure 90.58 miles east.

3. A ship sails from latitude 3° 52′ S. to latitude 4° 30' N., the course being N. W. by W. W.: required the distance and departure.

Ans. Dist. 1065 miles; dep. 939.2 miles W.

4. Two points are under the same meridian, one in latitude 52° 30′ N., the other in latitude 47° 10' N. A ship from the southern place sails due east, at the rate of 9 miles an hour, and two days after meets a sloop that had sailed from the other: required the sloop's direct course, and distance run.

Ans. Course S. 53° 28′ E.; dist. 537.6 miles.

5. If a ship from latitude 48° 27' S., sail S. W. by W. 7 miles an hour, in what time will she reach the parallel of 50° south? Ans. 23.914 hours.

SECTION III.

OF TRAVERSE SAILING.

15. When a ship, in going from one place to another, sails on different courses, it is called Traverse Sailing. The determination of the distance and course, from the place of departure to the place of termination, is called compounding or working the traverse. This is done by the aid of the "Traverse Table," which has already been explained, and the method of working the traverse, is in all respects similar to that adopted in the Prob. of Art. 34, page 123.

EXAMPLES.

1. A ship from Cape Clear, in lat. 51° 25′ N., sails, 1st, S. S. E. E. 16 miles; 2d, E. S. E. 23 miles; 3d. S. W. by W. W. 36 miles; 4th, W. & N. 12 miles; 5th, S. E. by E. E. 41 miles required the distance run, the direct

We first form the table below, in which we enter the courses, from the table of rhumbs, omitting the seconds, and then enter the latitudes and departures, taken from the traverse table, to the nearest quarter degree. Thus, in taking the latitude and departure for 25° 18' we take for 251°. The dif ference of latitudes gives the line AG, and the difference of departures the line GF

=

Difference of latitude 59.66 miles 1° 00' S.

Then, by formulas for the solution of right-angled tri

angles, we have,

As AG, diff. lat. ar. c. 8.224317 As sin course ar. c. .504995

: departure

:: radius,

19.64 1.293141: radius

10.000000

10.000000 departure 19.64 1.293141

: tang course 18° 13' 9.517458: distance 62.83 1.798136

Therefore the direct course is S. 18° 13′ E., and the distance 62.83 miles.

OF PLOTTING.

16. There is yet another method of finding the direct course and distance, much practiced by seamen, although it does not afford a high degree of accuracy. It is a method by plotting, which requires the use of a mariner's scale and a pair of dividers.

One of the scales marked on the mariner's scale, is a scale of chords, commonly called a scale of rhumbs, being divided to every quarter point of the compass; and there is also a second scale of chords divided to degrees. Both of these scales are constructed in reference to the same common radius, so that the chords on the scale of rhumbs correspond to those on the scale of marked chords. The manner of using the scales will appear in plotting the last example.

To construct this traverse, describe a circle with a radius equal to the chord of 60° and draw the meridian NS. Then take from the line of rhumbs the chord of the first course 2 points, and apply it from S to 1, to the right of NS, since the course is southeasterly, and draw Al; take, in like manner, the chord of the second course, 6 points, from S to 2, and lay it off also to the right of the meridian line. Apply the chord of the third course, 5 points, from S to 3, to the left of the meridian; the fourth course, 7 points from N to 4, to the left of NS, this course being northwesterly; and, lastly, apply the chord of the fifth course, 5 points, from S to 5, to the right of NS, and join all the lines as in the figure.

14

In the direction A1, lay off the distance AH= 16 miles from a scale of equal parts, and through the extremity H, draw HC parallel to A2, and lay off HC = 23 miles. Draw CD parallel to A3, and lay off CD36 miles; then draw DE parallel to A4, and lay off 12 miles; and lastly, draw EF parallel to A5, and lay off 41 miles, and F will be the place of the ship. Hence, we conclude that AF is the distance made good, and GAF is the course.

Applying, then, the distance AF to the scale of equal parts, we find it equal to 622 miles; and applying the chord Sa to the scale of chords, we find the course GAF = 181°.

2. A ship sails from a place in latitude 24° 32′ N., and runs the following courses and distances, viz., 1st, S. W. by W. dist. 45 miles; 2d, E. S. E. dist. 50 miles; 3d, S. W. dist. 30 miles; 4th, S. E. by E. dist. 60 miles; 5th, S. W. by S. W. dist. 63 miles required her latitude, and the direct course and distance from the place left to the place arrived at, and the construction of the traverse.

3. A ship from lat. 28° 32′ N. has run the following courses, viz., 1st, N. W. by N. 20 miles; 2d, S. W. 40 miles; 3d, N. E. by E. 60 miles; 4th, S. E. 55 miles; 5th, W. by S. 41 miles; 6th, E. N. E. 66 miles: required her latitude, the distance made good, and the direct course, also the construction of the traverse.

Ans. Dist. 70.2 miles, course E.

4. A ship from lat. 41° 12′ N. sails S. W. by W. 21 miles; S. W. S. 31 miles; W. S. W. 1 S. 16 miles; S.

E. 18 miles; S. W. W. 14 miles; then W. N. 30 miles required the latitude, the direct course, and the distance.

Ans.

Lat. 40° 05', course S. 52° 49′ W.
Dist. 111.7 miles.

5. A ship runs the following courses, viz.:

1st, S. E. 40 miles; 2d, N. E. 28 miles; 3d, S W. by